Higher Order Nonlinear Differential Equations Research Articles

Exact Analytical Solutions of Generalized Fifth-Order KdV Equation by the Extended Complex Method

The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

Solving Non Linear Differential Equations By Using A G - Homotopy Analysis Method

This paper proposes an efficient hybrid analytical method named the Generalized Homotopy Analysis Method (GHAM) for solving nonlinear differential equations. The proposed hybrid method consists of topology based Homotopy Analysis Method (HAM) and Laplace typed integral transform (G-transform). Compared with HAM, GHAM does not require the effort to perform repeated integration and differentiation, when it is applied to solve higher order differential equations. Compared with G-transform, GHAM serves as an effective approach to tackle higher-order nonlinear differential equations. The effectiveness of GHAM is illustrated through the analysis of numerical examples, and the obtained results are graphically depicted. Also, GHAM is effectively utilized to analyze the density of the forest cover incorporating various parameters, which include the seed reproduction, seed deposition, seed establishment rates, old trees due to aging and the coefficients of mortality due to space variables.

On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

The well-known first-order nonlinear difference equation yn+1=2yn−xyn2,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ y_{n+1}=2y_{n}-xy_{n}^{2}, \\quad n\\in {\\mathbb {N}}_{0}, $$\\end{document} naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

Oscillation Results for Nonlinear Higher-Order Differential Equations with Delay Term

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained in the literature. Some examples are presented to demonstrate the main results.

More Effective Conditions for Oscillatory Properties of Differential Equations

In this work, we present several oscillation criteria for higher-order nonlinear delay differential equation with middle term. Our approach is based on the use of Riccati substitution, the integral averaging technique and the comparison technique. The symmetry contributes to deciding the right way to study oscillation of solutions of this equations. Our results unify and improve some known results for differential equations with middle term. Some illustrative examples are provided.

Existence of Multiple Anti-Periodic Solutions for a Higher Order Nonlinear Difference Equation

Existence of Multiple Anti-Periodic Solutions for a Higher Order Nonlinear Difference Equation

Reproducing kernel functions and homogenizing transforms

A lot of problems of the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution and its effectiveness are as important as the modelling. In this paper, on the basis of reproducing kernel theory, the reproducing kernel functions have been obtained for solving some non-linear higher order differential equations. Additionally, for each problem the homogenizing transforms have been obtained.

Heat and mass transfer in electrically conducting micropolar fluid flow between two stretchable disks

In this research, the simultaneous effect of heat and mass transfer of a micropolar electrically conducting fluid which is confined between two stretchable disks is studied analytically. The obtained higher-order nonlinear partial differential equations are reduced into a set of nonlinear ordinary differential equations after using suitable similarity transformation. A well-recognized analytical method named Homotopy Perturbation Method is employed to solve nonlinear ordinary differential equations. The influence of Reynolds number, magnetic field, Eckert number, Prandtl number, Radiation parameter, and Schmidt number on velocity, microrotation, temperature, and concentration profiles are discussed and presented graphically. A comparison between outcomes obtained by the homotopy perturbation method (HPM) and the results of previously published work show the correctness and accuracy of the analytical method HPM. The coefficient of skin friction, couple stress, rate of heat, and mass transfer are also tabulated and compared with literature. The Maple software package is used for numerical computation.

Asymptotic behavior of solutions to difference equations in Banach spaces

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

Existence of Nonoscillation Solutions of Higher-Order Nonlinear Neutral Differential Equations

In this paper, we consider the following higher-order nonlinear neutral differential equations: dn dtn [x(t) + cx(t - τ)] + (-1)n+1[P(t)f1 (x(t - σ)) - Q(t)f2 (x(t - δ))] = 0; t ≥ t0 where τ; σ; δ ∈ R+, c ∈ R; c ̸= ±1, and P(t); Q(t) ∈ C([t0; ∞); R+), fi(u) ∈ C(R; R), ufi(u) > 0. we obtain the results which are some sufficient conditions for existence of nonoscillation solutions, special case of the equation has also been studied.

OSCILLATORY BEHAVIOR OF HIGHER ORDER NONLINEAR DIFFERENCE EQUATIONS

The authors present some new oscillation criteria for higher order nonlinear difference equations with nonnegative real coefficients of the form ... Both of the cases n even and n odd are considered. They give examples to illustrate their results.

Solvability of a system of higher order nonlinear difference equations

In this paper we show that the system of difference equations\[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

Global N-peakon weak solutions to a family of nonlinear equations

We use a double mollification method to study a family of high order nonlinear partial differential equations with peakon weak solutions. This method gives the approximated N-peakon weak solutions which satisfy weak consistent property. By a limiting process, we prove the global existence of N-peakon weak solutions. Moreover, the traveling speeds of single peakon weak solutions (soliton waves) are obtained directly by the limit of the mollification.

Oscillation criteria for higher-order neutral type difference equations

In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form ∆ rn ∆k−1 yn + pnyτn + qnf yσn = 0, n ≥ n0, 0.1 where rn, pn, qn ∈ [n0, ∞ , rn > 0, qn > 0; 0 ≤ pn ≤ p0 0; τσ = στ ; f u u ≥ m > 0 for u = 0. Moreover, we provide some examples to illustrate our main results.

The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations

This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials. Such characteristic makes of LPs a handy and powerful tool to approximate different kind of differential equations including: singular problems, initial condition and boundary-valued problems, equations with discontinuities, coupled differential equations, high-order equations, among others. Additionally, we show how the process to obtain the polynomials is straightforward and simple to implement; generating a compact, and easy to compute, expression. Even more, we present the process to approximate Gelfand's equation, an equation of an isothermal reaction, a model for chronic myelogenous leukemia, Thomas-Fermi equation, and a high order nonlinear differential equations with discontinuities getting, as result, accurate, fast and compact approximate solutions. In addition, we present the computational convergence and error studies for LPs resulting convergent polynomials and error tendency to zero as the order of LPs increases for all study cases. Finally, a study of CPU time shows that LPs require a few nano-seconds to be evaluated, which makes them suitable for intensive computing applications.

Optical soliton solutions of perturbing time-fractional nonlinear Schrödinger equations

This article presents a technique for finding solitary wave solutions of higher-order nonlinear differential equations. Time fractional nonlinear differential equations that have hierarchy contain higher orders are presented for the description of highly dispersive optical solutions. We find optical solitons with high dispersion order for time-fractional nonlinear perturbed Schrödinger equations of the fourth and sixth orders. Some figures are investigated to show the accuracy of the solutions obtained.

Method for finding highly dispersive optical solitons of nonlinear differential equations

A method for finding solitary wave solutions to nonlinear differential equations is presented. A generalization for the logistic function to obtain a solitary wave solution is introduced. Properties of this basic function are discussed. An algorithm for finding exact solutions in the form of a solitary wave for nonlinear differential equations is formulated. The method has significant advantages over other approaches of this type. Its advantage is due to the fact that in the calculations we do not use the form of a specific function. Our approach is particularly effective in finding exact solutions to high-order nonlinear differential equations used in describing the propagation of pulses in an optical fiber. The application of the exact solutions search method for finding highly dispersed solitons of nonlinear differential equations is shown.

On quasi-periodic solutions of forced higher order nonlinear difference equations

On quasi-periodic solutions of forced higher order nonlinear difference equations

Numerical solution of micropolar nanofluids with Soret, Dufor effects and multiple slip conditions

The objective of this study is to explore the effects of Dufour and Soret with multiple slip conditions on 3D boundary layer flow of a micropolar nanofluid. The mathematical modeling for the flow problem has been created. Using appropriate similarity transformation and non-dimensional variable, the governing non-linear boundary value problem is reduced to a coupled high order non-linear ordinary differential equations which is numerically solved. The solutions were using method bvp4c from matlab software for different quantities of governing parameters. The influences of different parameters on skin friction coefficients wall duo stress coefficients the Nusselt number Sherwood number as well as the velocities, temperature, and concentration are examined and discussed through tables and graphs. The finding results indicate the skin friction coefficients, wall duo stresses coefficients (along both x- and y-axes), the Nusselt number and the Sherwood number demonstrate decreasing behavior for an increase in slip parameters but the opposite effects are detected with the values of stretching ratio parameters. A comparison with previous studies available in the literature has been done and a very good agreement is obtained.

Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations

Hierarchy of the perturbed nonlinear Schrödinger equations is considered. Nonlinear differential equations of this hierarchy contain higher orders and can be used for description of highly dispersive optical solutions. A new approach for finding solitary wave solutions of high-order nonlinear differential equations is presented. This approach allows us to significantly simplify symbolic calculations. The main idea of the method is that we use expressions of the dependent variable and its derivatives in the differential equation the polynomial form of the solitary wave. We find optical solitons with high dispersion order for nonlinear perturbed Schrodinger equations of the fourth, sixth, eighth, tenth and twelfth orders.